5.74 TIME-DEPENDENT QUANTUM MECHANICS

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1 p. 1 MIT Deprtet of Cheistry 5.74, Sprig 005: Itroductory Qutu Mechics II Istructor: Professor Adrei Tooff 5.74 TIME-DEPENDENT QUANTUM MECHANICS The tie evolutio of the stte of syste is described by the tie-depedet Schrödiger equtio (TDSE): i ψ( r, t )= Ĥ ψ( r, t) t Most of wht you hve previously covered is tie-idepedet qutu echics, where we e tht Ĥ is ssued to be idepedet of tie: Ĥ = Ĥ (r). We the ssue solutio of the for: ψ(r, t)= ϕ(r )T(t) 1 ˆ ()ϕ r i Tt ()= H r Tt () t ϕ() r Here the left-hd side is fuctio of t oly, d the right-hd side is fuctio of r oly. This c oly be stisfied if both sides re equl to the se costt, E ˆ( )ϕ () r = E H ˆ r ()ϕ ()= r Eϕ () r () H r ϕ r Tie-Idepedet Schrödiger Eq. Here we ote tht H = ψ H ψ = E, so H is the opertor correspodig to E d drwig o clssicl echics we ssocite Secod eq.: H with the expecttio vlue of the eergy of the syste. i 1 T Tt () t = E t + ie Tt ()= 0 Solutio: Tt ()= Aexp( iet / )= Aexp( iωt)

2 p. So, for set of eigevectors ϕ ()with r correspodig eigevlues E, there re set of correspodig eigesolutios to the TDSE. The probbility desity ψ ( r, t)= ϕ (r) exp( iω t) ω = E / ( ) P = ψ (r, t)ψ(r, t)dr = ψ(r, t)ψ r, t is costt. P is idepedet of tie for the eigefuctios ψ ( r,t ). Therefore, ϕ ()re r clled sttiory sttes. However, ore geerlly syste y be represeted s lier cobitio of eigesttes: ( )= c ψ (r, t) = c e iω t ϕ (r) ψ r, t For such cse, the probbility desity will oscillte with tie: coherece. e.g., two eigesttes 1 ψ (r,t) = c ϕ e + c ϕ e iω t iω t 1 1 p t ()= ψ = ψ 1 + ψ = ψ 1 + ψ + Re ψ1 ψ = c ϕ + c ϕ + c c ϕ ϕ e i(ω ω t) + c 1 +i(ω ω 1)t c 1ϕϕ 1 e probbility desity oscilltes s cos(ω ω )t 1 This is siple exple of coherece (coheret superpositio stte). Icludig oetu ( wvevector) of prticle leds to wvepcet.

3 p. 3 TIME EVOLUTION OPERATOR More geerlly, we wt to uderstd how the wvefuctio evolves with tie. The TDSE is lier i tie. Sice the TDSE is deteriistic, we will defie opertor tht describes the dyics of the syste: For the tie-idepedet Hiltoi: ψ (t) =U( t,t 0 )ψ(t 0 ) ( )+ ih ψ r, t ψ ( r, t )= 0 (1) t To solve this, we will defie opertor T = exp( iht / ), which is fuctio of opertor. A fuctio of opertor is defied through its expsio i Tylor series: 1 iht T = exp[ iht ] = 1 iht +! = f( H) A iportt ote bout fuctios of opertor A : Give set of eigevlues d eigevectors of A, i.e., Aϕ = ϕ, you c show by ˆ ( )ϕ expdig the fuctio s polyoil tht f (A)ϕ = f 1 You c lso cofir fro the expsio tht T Multiplyig eq. 1 fro the left by T 1 we hve: = exp(iht / ), otig tht H is Hereti. t iht exp ( ) = 0 ψ r, t Now itegrtig t 0 t iht 0 exp ψ (r, t ) exp iht 0 ψ (r, t ) = 0

4 p. 4 ψ (r,t) = exp H t = U t,t )ψ (r,t ( 0 ( t 0 ) ψ (r,t 0 ) 0 ) E (t t 0 )/ ψ ( ) ψ ( r, t)= e r, t 0 or, ltertively, if we substitute the projectio opertor (idetity reltioship) ih ( t t 0 )/ ( t 0 ) = e ϕ ϕ ω i (t t 0 ) = e ϕ ϕ U t, ω E iht / ie t where I hve used the defiitio of the expoetil opertor for e ϕ = e ϕ. This for is useful whe ϕ re chrcterized; we ll develop U( t, t 0 ) ore lter. So ow we c write our tie-developig wve-fuctio s ω i (t t 0 ) ψ ( r,t ) = ϕ e ϕ ψ ( r,t 0 ) ω i (t t 0 ) c ( ) =ϕ e t 0 Tie-evolutio of coupled two-level syste (LS) It is coo to reduce or p proble oto LS. We the discrd the reiig degrees of freedo, or icorporte the s het bth, H= H 0 + H bth. Let s discuss the tie-evolutio of LS with tie-idepedet Hiltoi. Cosider LS with two (uperturbed) eigesttes ϕ d ϕ b with eigeeergies ε d ε b, which re the coupled through iterctio V b. H = ε + b ε b + V b b + b V b b ϕ ε V ε + ε V b ϕ b ε b = V b ε b ε

5 p. 5 Sice the Hiltoi is Hereti, (H ij = H ji ), we suggest If we defie the vribles V b = V b = Ve φ i iφ ε Ve H = i Ve +φ ε b ε + ε E = b ε ε = b The we c solve for the eigevlues of the coupled syste by solvig the seculr equtio det ( H λi ) = 0, givig ε ± = E ± + V Becuse the expressios get essy, we do t choose to fid the eigevectors for the coupled syste, ϕ ±, usig this expressio. Rther, we use substitutio where we defie: V t θ = 1 t θe H E I + t θe +φ 1 = i iφ θ (0 < θ < π/) V We ow fid tht we c express the eigevlues s ε ± = E ± secθ We ow wt to fid the eigesttes of the Hiltoi, ϕ ±, H ϕ ± = ε ± ϕ ± where e.g. ϕ + = c ϕ + c b ϕ b : ϕ =cos θ e φ si θ e φ + ϕ + i / i / ϕ b i / ϕ = si θ e φ cos θ e φ ϕ + i / ϕ b Orthoorl coplete + orthogol: ϕ ϕ + ϕb ϕ = 1 b

6 p. 6 Exie the liits: () We couplig (V/ << 1). Here θ 0, d ϕ + correspods to ϕ perturbed by the V b iterctio. ϕ correspods to ϕ b. (For θ 0 ϕ + ϕ ; ϕ ϕ b ) (b) Strog couplig (V/ >> 1). Now θ = 45º, d the /b bsis sttes re idistiguishble. The eigesttes re syetric d tisyetric cobitios: ~ 1 ( ϕ ± ϕ ) ϕ ± b Whether + or correspods to the syetric or syetric cobitio depeds o whether V is positive or egtive. (For V >>, θ = 45 ) We c scheticlly represet the eergies of these sttes: ε Ε ε + ε ε - ε b These eigesttes exhibit voided crossig.

7 p. 7 The tie-evolutio of this syste is give by our tie-evolutio opertor. ( U t,t e iω + (t t 0 ) ϕ e iω (t t 0 ) 0 ) = ϕ + ϕ + + ϕ ω ± = ε ± Now ϕ d ϕ b re ot the eigesttes preprig ϕ will led to tie-evolutio! Let s prepre the syste so tht it is iitilly i stte ϕ. (t 0 = 0) ψ (0) = ϕ Wht is the probbility tht it is foud i stte ϕ b t tie t? b ϕ b U t,t 0 ) P t t = ϕ b ( )= ϕ ψ () ( To evlute this, you eed to ow the trsfortio fro the ϕ,b to the ϕ ± bsis, give bove. This gives: V Pb ( t )= V + si Ω Rt 1 where the Rbi Frequecy Ω R = + V Ω R represets the frequecy t which probbility plitude oscilltes betwee ϕ d ϕ b sttes. V b () V + P t 0 t = π/ωr t Notice for V 0 ϕ ± ϕ,b (the sttiory sttes), d there is o tie-depedece. V π π For V >>, the Ω R = d P =1 fter t = =. Ω R V

8 p. 8 TIME-INDEPENDENT HAMILTONIAN There re two types of vlues tht we ofte clculte: Correltio plitude: Ct () = βϕ( t) esures the reseblce betwee the stte of your syste t tie t d trget stte β. The probbility plitude Pt) ( = Ct ( ) for set of eigesttes ϕ () ψ () ( ψ (0) C t = β t = β U t, t 0 ) j e ω jt = c j c,,j = c c e ω i t Expecttio vlues: At () = ψ t ()Aψ(t) ω t ψ () t = e c ϕ = c (t) ϕ ψ () t = e iω t c ϕ E E iω t ω = At () = cc e ϕ Aϕ, ω ω = c t, c t A () ()

9 p. 9 DENSITY MATRIX For syste described by wvefuctio ψ (t ) = c (t ) we showed At () = ψ (t) Aψ (t) () () = c t c t A We will ofte fid it useful to defie desity opertor, ρ () t ψ (t) ψ (t) = c t c t, ( ) () = ρ () t ( by defiitio ) ρ re the desity trix eleets. Substitutig, we see tht, (, At ) = A ρ (t ) = Tr A ρ () t Trce Properties: 1) cyclic ivrice Tr ( ABC ) = Tr ( CAB ) = Tr (BCA) ) ivrit to uitry trsfortio Tr ( S AS ) = Tr ( A ) Pure vs. Mixed Sttes Why would we eed the desity trix? It helps for ixed sttes. 1) pure sttes: syste chrcterized by wvefuctio (previous pge) ) ixed sttes: ot chrcterized by sigle wvefuctio > sttisticl ixtures eseble t therl equilibriu > idepedetly prepred sttes > o phse reltioship betwee eleets of ixture

10 p. 10 For eseble of systes with probbility p of occupyig qutu stte p = 1 Properties: A() t = p ψ (t) A ψ (t) ρ( t) p ψ () t ψ () t A() t = Tr Aρ( t ) ψ, with 1) ρ is Hereti ρ = ρ ) Tr(ρ)=1 Norliztio 3) Tr ( ρ )=1 for pure stte < 1 for ixed stte Let s loo t the desity trix eleets for ixture: where ρ = ρ = p ψ ψ ϕ = c c : expsio coefficiet for eigestte of wvefuctio = p c ( c ) = c c coefficiets for eigestte verged over ixture Digol eleets ( = ) ρ = p c =c c = p probbility of fidig syste i ixture i stte POPULATION ( 0)

11 p. 11 Off-Digol Eleets ( ) coplex hve phse fctor describe the evolutio of coheret superpositios. COHERENCES For rbitrry stte χ, the expecttio vlue of the desity trix: χ ρχ gives the totl probbility of fidig prticle i the pure stte χ withi the ixture. Defiitio of the desity trix We will soeties refer to the desity trix t therl equilibriu ρ 0 ( or ρ eq ), which is chrcterized by therlly distributed popultios i the qutu sttes βe ρ = p = e Z where Z is the prtitio fuctio. More geerlly, the desity trix c be defied s βh ρ= e Z where Z = Tr(e βh ). For H = E, ρ = e βh βe = e δ

12 p. 1 TIME-EVOLUTION OF DENSITY MATRIX Follows turlly fro defiitio of ρ d T.D.S.E. i H ψ ψ = ψ H (H = H) i ψ = t t ρ = ψ ψ = ψ ψ + ψ ψ t t t t i i = H ψ ψ + H ψ ψ ρ i = [H, ρ] t Liouville-Vo Neu Eq. For tie-idepedet Hiltoi: ρ ()= t ρ(t) = ψ (t) ψ (t) ψ () t = U ( t, t 0 ) ψ (t 0 ) = i ω (t t 0 ) e ψ (t 0 ) ω t (t t 0 ) ρ ()= e ψ ( t 0 ) ψ ( t 0 ) e i +ω i (t t 0 ) i = e ω (t t ) 0 ρ ( t 0 ) ω = E E Popultios: ρ ()= t ρ (t 0 ) tie-ivrit Cohereces: oscillte t eergy splittig ω

13 p. 13 APPENDIX: PROPERTIES OF OPERATORS 1 1) The iverse of  (writte  ) is defied by ˆ 1 ˆ = ˆ ˆ A 1 A AA = Î ) The trspose of  (writte AT ) is T ( A ) = A q q If A T = A the the trix is tisyetric 3) The trce of  is defied s Tr  ()= A qq q 4) Hereti Adjoit of A ˆ (writte A ˆ ) A = ( A ) ( A ˆ ) = ˆ q ( A q ) ˆ ˆ T 5)  is Hereti if  =  ( Aˆ T ) = A If A ˆ is Hereti, the A ˆ is Hereti. 6)  is uitry opertor if: ˆ ˆ 1 A = A ( A ˆ T ) = A ˆ 1 ) q ˆˆ AA =1 ( AA ˆˆ = δ q